Determine the precision and accuracy of these data for warfarin. Sample 1 Sample 2 Sample 3 Known concentration (ng/mL): 24.2 75.9 291 Experimentally determined values (ng/mL): 35.5 70.3 303 20.6 56.5 283 25.9 69.1 261 22.7 88.4 307 22.6 68.0 253 26.8 71.1 279

answer:

ta da!

questions arthur t. benjamindec. 30, 2009

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arthur t. benjamin has two passions: magic and math. when not amazing audiences around the country — squaring five-digit numbers in his head or guessing your number, any number — the mathemagician is a professor of math at harvey mudd college in claremont, calif. there’s even whimsy to his ph.d. dissertation, at johns hopkins, titled “turnpike structures for optimal maneuvers”: the maneuvers were inspired by a way of arranging chinese checkers to move expeditiously across the board. below, dr. benjamin shares some of the concepts from his dvd course “the joy of mathematics” and his book “secrets of mental math: the mathemagician’s guide to lightning calculation and amazing math tricks.”

follow the instructions below. i bet i can predict your answers. how do i do it? explanations follow.

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1. choose a number from 1 to 70 and then divide it by 7. (i’ll be nice and let you use a calculator, but you’ll need one that has at least seven decimal places.) if your total is a whole number (that is, no digits after the decimal point) divide the answer by 7 again. is there a 1 somewhere after the decimal point? i predict that the number after the 1 is 4. am i right? now add up the first six digits after the decimal point.

your answer is 27.

how do i do it?

the six fractions 1/7 to 6/7 have the same repeating sequence of six numbers after the decimal point, each fraction starting with a different number in the sequence 142857. think of these numbers as in a circle (diagram a), and going around that circle forever.

the first number after the decimal point in the decimal version of 1/7 is 1 (.142857); of 2/7, 2 (.285714); of 3/7, 4 (.428571); of 4/7, 5 (.571428); of 5/7, 7 (.714285); and of 6/7, 8 (.857142). adding 1, 4, 2, 8, 5 and 7, you always get 27

hence, when subtracting the smaller number from the larger one, the identical remainders will cancel each other out, and the result is — poof! — a multiple of 9.

algebraically, what you have is (9x + r) - (9y + r) = 9(x - y).

second, every multiple of 9 (9, 18, 27, 36, etc.) has the property that its digits will always add up to a multiple of 9. for example, the number 4,968 is a multiple of 9 (552 × 9), and its digits add up to 27 (3 × 9).

because of how the problem is stated, with four-digit numbers, the sum of the digits will always be 9 or 18 or 27. adding 1 + 8 or adding 2 + 7 (as specified in the problem) again gives 9.

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6. in the highlighted grid on the calendar (diagram c), circle four dates so there is one circled in each row and each column. add the four circled numbers.

your total is 64.

how do i do it?

look at the sunday dates, to the left of the grid (diagram e).

now think of the numbers for monday through thursday as simply those sunday numbers plus 1, 2, 3 or 4.

photo

in choosing four numbers from different rows and columns, you are just adding the sunday numbers (54) plus 1 + 2 + 3 + 4 (10), which equals 64.

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7. in a table like the one shown in diagram d, write any number in rows 1 and 2. add those numbers, and put the total in row 3. add the numbers in rows 2 and 3, and put the answer in row 4. continue this process until you have numbers in all 10 rows. now add up the 10 numbers and divide by the number in row 7.

your answer is 11.

how do i do it?

suppose you start with the number x in row 1 and number y in row 2. then row 3 will be x + y, which leads to row 4 being x + 2y, and so on. diagram f shows what the final table looks like.

the grand total is 11 times 5x + 8y. note that 5x + 8y is in row 7. hence, dividing the total by what’s in row 7 will always yield the answer 11.

tip: to look like a human calculator, ask somebody to write numbers in the table, as before. when he shows you the list, make a great show of “mind reading” and challenge him to add all the numbers with a calculator faster than you can in your head. just multiply row 7 by 11card number. it is not the codabar. it is the luhn

step-by-step explanation: